Langlands Correspondence Research Articles

Geometric Langlands correspondence for , over the pair of pants

We establish the geometric Langlands correspondence for rank-one groups over the projective line with three points of tame ramification.

Twisted compactifications of 3d mathcal{N} = 4 theories and conformal blocks

Three-dimensional mathcal{N} = 4 supersymmetric quantum field theories admit two topological twists, the Rozansky-Witten twist and its mirror. Either twist can be used to define a supersymmetric compactification on a Riemann surface and a corresponding space of supersymmetric ground states. These spaces of ground states can play an interesting role in the Geometric Langlands program. We propose a description of these spaces as conformal blocks for certain non-unitary Vertex Operator Algebras and test our conjecture in some important examples. The two VOAs can be constructed respectively from a UV Lagrangian description of the mathcal{N} = 4 theory or of its mirror. We further conjecture that the VOAs associated to an mathcal{N} = 4 SQFT inherit properties of the theory which only emerge in the IR, such as enhanced global symmetries. Thus knowledge of the VOAs should allow one to compute the spaces of supersymmetric ground states for a theory coupled to supersymmetric background connections for the full symmetry group of the IR SCFT. In particular, we propose a conformal field theory description of the spaces of ground states for the T[SU(N)] theories. These theories play a role of S-duality kernel in maximally supersymmetric SU(N) gauge theory and thus the corresponding spaces of supersymmetric ground states should provide a kernel for the Geometric Langlands duality for special unitary groups.

Spectral Data for Spin Higgs Bundles

Abstract In this paper we determine the spectral data parametrizing Higgs bundles in a generic fiber of the Hitchin map for the case where the structure group is the special Clifford group with fixed Clifford norm. These are spin and “twisted” spin Higgs bundles. The method used relates variations in spectral data with respect to the Hecke transformations for orthogonal bundles introduced by Abe. The explicit description also recovers a result from the geometric Langlands program, which states that the fibers of the Hitchin map are the dual abelian varieties to the corresponding fibers of the moduli spaces of projective orthogonal Higgs bundles (in the even case) and projective symplectic Higgs bundles (in the odd case).

Relative Functoriality and Functional Equations via Trace Formulas

Langlands’ functoriality principle predicts deep relations between the local and automorphic spectra of different reductive groups. This has been generalized by the relative Langlands program to include spherical varieties, among which reductive groups are special cases. In the philosophy of Langlands’ “beyond endoscopy” program, these relations should be expressed as comparisons between different trace formulas, with the insertion of appropriate L-functions. The insertion of L-functions calls for one more goal to be achieved: the study of their functional equations via trace formulas. The goal of this article is to demonstrate this program through examples, indicating a local-to-global approach as in the project of endoscopy. Here, scalar transfer factors are replaced by “transfer operators” or “Hankel transforms” which are nice enough (typically, expressible in terms of usual Fourier transforms) that they can be used, in principle, to prove global comparisons (in the form of Poisson summation formulas). Some of these examples have already appeared in the literature; for others, the proofs will appear elsewhere.

Models of Representations and Langlands Functoriality

AbstractIn this article we explore the interplay between two generalizations of the Whittaker model, namely the Klyachko models and the degenerate Whittaker models, and two functorial constructions, namely base change and automorphic induction, for the class of unitarizable and ladder representations of the general linear groups.

A torsion Jacquet-Langlands correspondence

We prove a numerical form of a Jacquet–Langlands correspondence for torsion classes on arithmetic hyperbolic 3-manifolds.

Jordan blocks of cuspidal representations of symplectic groups

Let $G$ be a symplectic group over a nonarchimedean local field of characteristic zero and odd residual characteristic. Given an irreducible cuspidal representation of G, we determine its Langlands parameter (equivalently, its Jordan blocks in the language of Moeglin) in terms of the local data from which the representation is explicitly constructed, up to a possible unramified twist in each block of the parameter. We deduce a Ramification Theorem for $G$, giving a bijection between the set of endo-parameters for $G$ and the set of restrictions to wild inertia of discrete Langlands parameters for $G$, compatible with the local Langlands correspondence. The main tool consists in analysing the intertwining Hecke algebra of a good cover, in the sense of Bushnell--Kutzko, for parabolic induction from a cuspidal representation of $G\times\mathrm{GL}_n$, seen as a maximal Levi subgroup of a bigger symplectic group, in order to determine its (ir)reducibility; a criterion of Moeglin then relates this to Langlands parameters.

Outline of the proof of the geometric Langlands conjecture for $GL_2$

Outline of the proof of the geometric Langlands conjecture for $GL_2$

L-groups and the Langlands program for covering groups : a historical introduction

L-groups and the Langlands program for covering groups : a historical introduction

L-groups and the Langlands program for covering groups

L-groups and the Langlands program for covering groups

On the universal mod p supersingular quotients for GL2(F) over [formula omitted] for a general [formula omitted

Let F/Qp be a finite extension. We explore the universal supersingular mod p representations of GL2(F) by computing a basis for their spaces of invariants under the pro-p Iwahori subgroup. This generalizes works of Breuil and Schein (from Qp and the totally ramified cases to an arbitrary extension F/Qp). Using these results we then construct, for an unramified F/Qp, quotients of the universal supersingular modules which have as quotients all the supersingular representations of GL2(F) with a GL2(OF)-socle that is expected to appear in the mod p local Langlands correspondence. A construction in the case of an extension of Qp with inertia degree 2 and suitable ramification index is also presented.

Le nombre des systèmes locaux ℓ-adiques sur une courbe

Let X1 be a projective, smooth, and geometrically connected curve over Fq with q=pn elements where p is a prime number, and let X be its base change to an algebraic closure of Fq. The goal of this article is to announce a formula for the number of irreducible ℓ-adic local systems (ℓ≠p) with a fixed rank over X fixed by the Frobenius endomorphism. This number behaves like a Grothendieck–Lefschetz fixed point formula for a variety over Fq, which generalises a result of Drinfeld in rank 2 and proves a conjecture of Deligne. We also sketch our method, which consists in passing to the automorphic side by Langlands correspondence, then calculating all the terms in Arthur's non-invariant trace formula and linking the geometric part of trace formula to the number of Fq-points of the moduli space of stable Higgs bundles.

On an invariant bilinear form on the space of automorphic forms via asymptotics

This article concerns the study of a new invariant bilinear form $\mathcal B$ on the space of automorphic forms of a split reductive group $G$ over a function field. We define $\mathcal B$ using the asymptotics maps from Bezrukavnikov-Kazhdan and Sakellaridis-Venkatesh, which involve the geometry of the wonderful compactification of $G$. We show that $\mathcal B$ is naturally related to miraculous duality in the geometric Langlands program through the functions-sheaves dictionary. In the proof, we highlight the connection between the classical non-Archimedean Gindikin-Karpelevich formula and certain factorization algebras acting on geometric Eisenstein series. We then give another definition of $\mathcal B$ using the constant term operator and the inverse of the standard intertwining operator. The form $\mathcal B$ defines an invertible operator $L$ from the space of compactly supported automorphic forms to a new space of "pseudo-compactly" supported automorphic forms. We give a formula for $L^{-1}$ in terms of pseudo-Eisenstein series and constant term operators which suggests that $L^{-1}$ is an analog of the Aubert-Zelevinsky involution.

On the Chebotarev–Sato–Tate phenomenon

We study the Chebotarev–Sato–Tate phenomenon that concerns the distribution of Artin symbols and Frobenius angles. From the recent work of Barnet-Lamb et al., we derive some unconditional results in the case of Hilbert modular forms. Furthermore, under the Langlands functoriality conjecture and the generalised Riemann hypothesis, we give two effective versions of such distributions, which present a modular variant of the results of Bucur, Kedlaya, and V.K. Murty.

Affinoids in the Lubin–Tate Perfectoid Space and Simple Supercuspidal Representations I: Tame Case

We construct a family of affinoids in the Lubin-Tate perfectoid space and their formal models such that the middle cohomology of the reductions of the formal models realizes the local Langlands correspondence and the local Jacquet-Langlands correspondence for simple supercuspidal representations in the case where the dimension of Galois representations is prime to the residue characteristic. The reductions of the formal models are isomorphic to the perfections of Artin-Schreier varieties associated to quadratic forms.

Local root numbers and spectrum of the local descents for orthogonal groups : p-adic case

We investigate the local descents for special orthogonal groups over p-adic local fields of characteristic zero, and obtain an explicit spectral decomposition of the local descents at the first occurrence index in terms of the local Langlands data via the explicit local Langlands correspondence and explicit calculations of relevant local root numbers. The main result can be regarded as a refinement of the local Gan-Gross-Prasad conjecture.

Arithmetic gauge theory: A brief introduction

Much of arithmetic geometry is concerned with the study of principal bundles. They occur prominently in the arithmetic of elliptic curves and, more recently, in the study of the Diophantine geometry of curves of higher genus. In particular, the geometry of moduli spaces of principal bundles holds the key to an effective version of Faltings’ theorem on finiteness of rational points on curves of genus at least 2. The study of arithmetic principal bundles includes the study of Galois representations, the structures linking motives to automorphic forms according to the Langlands program. In this paper, we give a brief introduction to the arithmetic geometry of principal bundles with emphasis on some elementary analogies between arithmetic moduli spaces and the constructions of quantum field theory.

Converse theorems and the local Langlands correspondence in families

We prove a descent criterion for certain families of smooth representations of {text {GL}}_n(F) (F a p-adic field) in terms of the gamma -factors of pairs constructed in Moss (Int Math Res Not 2016(16):4903–4936, 2016). We then use this descent criterion, together with a theory of gamma -factors for families of representations of the Weil group W_F (Helm and Moss in Deligne–Langlands gamma factors in families, arXiv:1510.08743v3, 2015), to prove a series of conjectures, due to the first author, that give a complete description of the center of the category of smooth W(k)[{text {GL}}_n(F)]-modules (the so-called “integral Bernstein center”) in terms of Galois theory and the local Langlands correspondence. An immediate consequence is the conjectural “local Langlands correspondence in families” of Emerton and Helm (Ann Sci Éc Norm Supér (4) 47(4):655–722, 2014).

On tensor product L-functions for quaternion unitary groups and GL(2) over totally real number fields: Mixed weight cases

We prove an algebraicity of special values of tensor product L-functions for quaternion unitary groups and GL(2) in the mixed weight cases over any totally real number field. As a consequence, we can prove a period relation for automorphic forms of quaternion unitary groups of degree 2 and GSp(4) when these automorphic forms correspond under the conjectural Jacquet–Langlands correspondence and a period relation for endoscopic liftings of GSp(4) in very general situations.

Cycles Cohomology and Geometrical Correspondences of Derived Categories to Field Equations

The integral geometry methods are the techniques could be the more naturally applied to study of the characterization of the moduli stacks and solution classes (represented cohomologically) obtained under the study of the kernels of the differential operators of the corresponding field theory equations to the space-time. Then through a functorial process a classification of differential operators is obtained through of the co-cycles spaces that are generalized Verma modules to the space-time, characterizing the solutions of the field equations. This extension can be given by a global Langlands correspondence between the Hecke sheaves category on an adequate moduli stack and the holomorphic bundles category with a special connection (Deligne connection). Using the classification theorem given by geometrical Langlands correspondences are given various examples on the information that the geometrical invariants and dualities give through moduli problems and Lie groups acting.